Questions

1)

Answer the following questions on thermal expansion.

(a)

A steel rod has a length of exactly 20 cm at $30~\rm ^o C$. How much longer is it at $50 ~\rm ^o C$? The coefficient of linear expansion of steel is $11\times 10^{-6}~\rm /^o C$. (5 points)

(b)

The coefficient of linear expansion of a certain solid substance is $\alpha$. Suppose that you made a square plate out of the substance of size $L\times L$. Then what is the coefficient of area expansion (i.e., $\Delta A/A \over \Delta T$ where $A$ is the area) of this plate? Prove your answer, neglecting the terms proportional to $\Delta T ^2$ or higher order terms of $\Delta T$. (10 points)

(c)

Normally, you can open a bottle of coke without spilling any of the liquid. But if you shake the bottle and immediately take the cap off (just to see if your cap is a winning cap), the coke will spew out everywhere even though you have not changed the temperature of the coke (from the act of shaking the bottle). Explain why this happens. (5 points)

2)

Answer the following questions on specific heat and latent heat.

(a)

A metal of mass $m_x$ is heated to a temperature $T_x$ and then dropped into a container containing a liquid of mass $m_l$ initially at temperature $T_l$, where $T_x > T_l$. If the final equilibrium temperature is $T_f$, find the specific heat of the metal. The specific heat of the liquid is $c_l$. (5 points)

(b)

Silver has (i) specific heat of $236~\rm J/kg\cdot K$, (ii) latent heat of fusion $105 ~\rm kJ/kg$, and (iii) latent heat of vapourisation $2336~\rm kJ/kg$. The melting point of silver is at 1235 K and the boiling point at 2323 K. Given this information, calculate the minimum amount of heat, in joules, required to completely melt 130 g of silver initially at $15 ~\rm ^o C$. Assume that the specific heat does not change with temperature. (10 points)

(c)

The internal energy of $n ~\rm mol$ of a certain gas can be expressed in terms of temperature as $E_{\rm int} = 3nRT$. What is the $C_V$ (constant-volume molar specific heat) of this gas? (5 points)

3)

An ideal monatomic gas within a chamber undergoes the processes shown in the $P$-$V$ diagram of Fig. 1.

Figure 1: Prob 3

(a)

Calculate the net heat added to the system during one complete cycle. Give the correct sign of the heat. (5 points)

Calculate the work done by the gas and calculate the heat added to the gas during (b) process $B\rightarrow C$ (5 points), (c) process $C\rightarrow A$ (5 points), and (d) process $A\rightarrow B$ (5 points). Hint: The internal energy of an ideal monatomic gas is $E_{\rm int}={3\over 2}nRT$.

4)

A vertical cylinder with a heavy piston contains air at 300 K. The initial pressure is 200 kPa, and the initial volume is $0.350 ~\rm m^3$. Take the molar mass of air as $28.9~\rm g/mol$ and assume that $C_V =5R/2$.

(a)

Find the specific heat of air at constant volume in units of $\rm J/kg\cdot C$. (5 points)

(b)

What can you say about the number of degrees of freedom of each molecule of the air? Estimate the r.m.s. speed of the air molecules in the initial state and compare it to that of an ideal monatomic gas at the same temperature. Are they equal or not? (10 points)

(c)

Assume the condition of the initial state and that the heavy piston is free to move. Find the energy input required to raise temperature to 700 K. (5 points)

5)

Figure 2 shows $n$ mol of an ideal monatomic gas being taken through a reversible cycle that consists of two isothermal processes at temperatures $3T_1$ and $T_1$ and two constant-volume processes with volumes $V_1$ and $4V_1$.

Figure 2: Prob 5

(a)

What is the efficiency of the cycle? Give the derivation. (15 points)

(b)

What is the total entropy change in one cycle? In realistic engines with irreversible processes, what can you say about the total entropy change in the whole system (engine + surroundings)? (5 points)